I can't start from the Greek original but, as a translated piece, the first of these seems a bit mechanical, and I prefer the second for style (it's a little shorter, too).
This edit has been a long time in coming, but for what it's worth:
An opinion on "the quality of the Latin" must take into account various ideas of what that might mean. A principal idea that used to be taught (and may be still) is that good classical prose seldom allowed a change of subject within a whole period — such changes, if felt to be unavoidable, perhaps being made by the use of colons or semi-colons. Other points included careful consideration of the word order, including how emphasis might be introduced and ambiguity avoided. Some consistency in the style of syntax was always looked for (this, which generated whole books on prose composition, would require too long a passage of explanation to give here). And simplicity was held to be desirable, to preclude the difficulty in construing that is always to be avoided : for example, a complex series of adverbial clauses within a single period was seldom a good idea. Based on these inculcated principles, my initial instinct on reading through these pieces was to prefer (A).
For a more considered analysis, I thought I might do well to use as a guide Rawlinson's 1858 translation from the Greek, direct into English. To identify separate subjects for each period, I began by arbitrarily editing the matter into eight 'propositions', as follows (ideally, I would have expected eight sentences to correspond roughly to these) :
(i) Persians say Phoenicians began the quarrel. (ii) Phoenicians migrated from Red to Mediterranean Sea (where they remain). (iii) At once, they began long mercantile voyages with wares from Egypt and Assyria. (iv) They traded with inter alia Argos, principal city of (now) Hellas. (v) Business was almost complete after 5-6 days there. (vi) Women having approached, including Io, daughter of Inachus, they were set upon by Phoenicians. (vii) Most escaped but some, including Io , were carried off to Egypt, instigating the quarrel. (viii) The Persian version differs widely from the Phoenician.
Each of your translators makes a different 'grouping' of the propositions:
— Schweighäuser (S) has five sentences, of in all 157 words. S1 (11 words) comprises proposition (i) only; S2 (41) has (ii), (iii), (iv) and part of (v). S3 (16) has the end of (v). S4 (44) has (vi) and (vii). S5 (45) has only (viii);
— The alternative (A) has four sentences, in all 132 words. A1 (45 words) comprises propositions (i), (ii) and (iii). A2 (12) has, (iv), (v) and (vi), A3 (37)has (vii). A4 (38), like S5, has only (viii).
I have long forgotten what little Greek I ever learned, and can't comment on the original text of Herodotus and how well it is matched by the Latin. All things considered, I think that neither amounts to what is called 'fine writing' and certainly does not suggest the so-called golden age; but in spite of violating the principles (as both do )that I suggested above — except that of brevity in the case of (A) — Schweighäuser's text seems noticeably more clumsy.